Lines and angles

Chapter-6, Lines and Angles, 9 Grade CBSE

1. Lines and angles Chapter-6 For Grade -IX By Sri . Siddalingeshwara BP M.Sc , B.Ed , ( P.hd ) Contact: 9590700228 Email- [email protected]

2. CONTENTS: Introduction Basic Terms and Definition Intersecting Lines and Non-intersecting Lines Pairs of Angles Parallel Lines and a Transversal Line Lines Parallel to the same Line Angle Sum Property of a triangles S ummary

3. 1. Introduction: Geometry is the branch of mathematics which deals with shape, size, position, spatial relationships and properties of different figures. There are different parameter involved to define a shape. The two important parameters are lines and angles. In this article, we are going to discuss two important parameters called lines and angles in detail. The Presentation on lines and angles are given, which covers the various concepts such as parallel lines, transversal, angles, intersecting lines, interior angles are explained with the examples. Hence, The entire geometry begins with a point. A point is a dimensionless entity which specifies the location or position. It is represented using a dot symbol and its length is zero. Point Zero dimension To draw a line are to construct the geometrical diagram we need some reference t o draw a line the reference is called point.

4. 2.Basic Terms and Definition What is a point, line , line segment and Ray?

5. And some more basic term regarding to Lines and notation.

6. What is an angle? If a ray is rotated about its end point then the measure of its rotation between the final and initial position of the ray is known as an angle. In fig . is the initial position of the ray and when it is rotated about its point i.e . O, the final position is represented by ray . The measure of this rotation is measured in angles. The angle between the initial and final position of a ray is measured as ∠AOB.   Initial position of Ray Final position of Ray B A O Vertex OA and OB are the arms and they making angle at O called vertex.

7. In General:

8. Complementary Angle: Two angle whose sum is 90 Degree is called Complementary angle Supplementary Angle: Two angle whose sum is 180 degree are called supplementary angle

9. Vertically opposite Angle:

10. 3. Intersecting Lines and Non-intersecting Lines When two or more lines cross each other in a plane, they are called intersecting lines. The intersecting lines share a common point, which exists on all the intersecting lines, and is called the point of intersection Here, lines PQ and RS intersect at point O, which is the point of intersection. o Properties of intersecting lines 1.The intersecting lines (two or more) meet only at one point always. 2.The intersecting lines can cross each other at any angle. This angle formed is always greater than 0 degree and less than 180 degree. 3.Two intersecting lines form a pair of vertical angles. The vertical angles are opposite angles with a common (which is the point of intersection). Here, ∠a and ∠c are vertical angles and are equal. Also, ∠b and ∠d are vertical angles and equal to each other. ∠a + ∠d = straight angle = 180° Intersecting Lines

11. Non-intersecting Lines When more than two lines do not intersect with each other, they are termed as non-intersecting lines. Non-intersecting lines hold the following properties. 1.Non-intersecting lines never share any endpoint. 2.They never meet. 3.The distance between the two parallel non-intersecting lines is fixed. 4.In the case of two parallel lines, the perpendicular line drawn at any point to one line will be perpendicular to the other line as well. 5.The length of the common perpendicular between the two parallels at any location will always be the same. Observe the following figure of two non-intersecting, parallel lines ‘PQ' and ‘RS' with a perpendicular distance denoted by 'c' and 'd'. c d

12. 4. Pairs of Angles: You have learnt the definition of some of the pairs of angles such as Complementary Angles, supplementary angles, adjacent angles, linear pair of angles etc. Now, Let us find out the relation between the angles formed when a ray stands on a line.

13. This is about Linear pair of angles From Axiom 6.1 and 6.2

14. An axiom, postulate or assumption is a statement that is taken to be true What is Axiom? Axiom 6.1: If a ray stands on a line, then the sum of two adjacent angles so formed is 180 degree. Then they are called a Linear pair of angles. Axiom 6.2: If the sum of two adjacent angle is 180 degree , then the common arms of the angles form a line “The two axioms above together is called the Linear pair axioms” Now Let us study the Properties of Lines and angles using some Theorems A theorem is a statement that can be demonstrated to be true by accepted mathematical operations, Postulate /axioms and arguments What is Theorem? Steps Involved to write a Theorem 1.Read the Statement of the Theorem 2. Given 3. To proof 5 . Proof 6 . Conclusions 4.construction

15. “If two Lines Intersect each other, then the vertically opposite angles are equal” Statement: “If two Lines Intersect each other, then the vertically opposite angles are equal” Theorem 6.1:

16. EXERCISE 6.1:

17. 5. Parallel Lines and a Transversal Transversal line

18. Theorem 6.2:

19. Theorem 6.3:

20. Theorem 6.4:

21. Theorem 6.5 :

22. 6.Lines Parallel to the same Line

23. Theorem 6.6 :

24. Example 6:

25. 7.Angle Sum Property of a triangles: One common property about triangles is that all three interior angles add up to 180 degrees. This now brings us to an important theorem in geometry known as Triangle Angle Sum Theorem. According to the Triangle Angle Sum Theorem, the sum of the three interior angles in a triangle is always 180°

26. Example 1: Example2:

27. Theorem 6.7 :

28. Theorem 6.8:

29. Note: Angle sum property of interior and exterior angle of Triangle For Interior angle of Triangle For Exterior angle of triangle

30. Example 7:

31. EXERCISE 6.3: 5

32. 8.Summary

33. References: 1.NCERT Text book 2.Google Sources 3.Slide Share Thank you